\(\mathrm{y = (x - 3)^2 + 6}\)y = 42The graphs of the given equations in the xy-plane intersect at the...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{y = (x - 3)^2 + 6}\)
  • Second equation: \(\mathrm{y = 42}\)
  • Need to find where these graphs intersect
• What this tells us: At the intersection point, both equations have the same y-value for the same x-value

2. INFER the solution approach

• Since both expressions equal y, I can set them equal to each other
• This gives us one equation with one unknown (x) that I can solve

3. SIMPLIFY by setting up and solving the equation

Set the right sides equal:

\(\mathrm{(x - 3)^2 + 6 = 42}\)

Subtract 6 from both sides:

\(\mathrm{(x - 3)^2 = 36}\)

4. CONSIDER ALL CASES when taking the square root

Take the square root of both sides:

\(\mathrm{x - 3 = ±\sqrt{36}}\)

\(\mathrm{x - 3 = ±6}\)

This creates two separate equations:

• Case 1: \(\mathrm{x - 3 = 6}\), so \(\mathrm{x = 9}\)
• Case 2: \(\mathrm{x - 3 = -6}\), so \(\mathrm{x = -3}\)

5. APPLY CONSTRAINTS to select the available answer

• Both solutions (\(\mathrm{x = 9}\) and \(\mathrm{x = -3}\)) are mathematically valid
• Looking at the answer choices: A) -3, B) 3, C) 6, D) 42
• Only \(\mathrm{x = -3}\) appears among the choices

Answer: A) -3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students often only consider the positive square root when solving \(\mathrm{(x - 3)^2 = 36}\), finding \(\mathrm{x - 3 = 6}\) and thus \(\mathrm{x = 9}\). Since 9 isn't among the answer choices, this leads to confusion and guessing between the available options.

Second Most Common Error:

Poor TRANSLATE reasoning: Some students don't recognize that "intersection point" means setting the equations equal. Instead, they might try to substitute one equation into the other incorrectly or attempt to solve each equation separately. This leads to confusion about what they're actually solving for.

The Bottom Line:

This problem tests whether students understand both the concept of function intersections and the fundamental principle that square roots have two solutions. Missing either piece derails the entire solution process.